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<p>Let <i>f</i> be a normalized primitive cusp form of even integral weight for <i>Γ</i> = <i>SL</i>(2, ℤ), and let <i>g</i> be a normalized Hecke–Maass cusp form. In the present paper, for any prescribed integer <i>ℓ</i> ≥ 2, we intend to investigate the average estimates of the Fourier coefficients <i>λ</i><sub><i>f</i>⊗<i>f</i>⊗</sub>…<sub>⊗<i>ℓf</i>⊗<i>g</i></sub>(<i>n</i>) of the (<i>ℓ</i>+1)-fold product <i>L</i>-functions <i>L</i>(<i>f</i> ⊗ <i>f</i> ⊗…⊗<sub><i>ℓ</i></sub><i> f</i> ⊗ <i>g, s</i>) involving <i>f</i> and <i>g</i>, where <i>f</i> ⊗ <i>f</i> ⊗…⊗<sub><i>ℓ</i></sub><i> f</i> ⊗ <i>g</i> is the (<i>ℓ</i>+1)-fold product associated with <i>f</i> and <i>g</i>. As a direct application, we also derive quantitative results for the sign changes of the sequence {λ<sub><i>f</i>⊗<i>f</i>⊗</sub>…⊗<sub><i>ℓ f</i>⊗<i>g</i></sub>(<i>n</i>)}<sub><i>n</i>≥1</sub> in short intervals, with the indices supported at the positive integers and certain binary quadratic forms.</p>

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The asymptotic distributions of Fourier coefficients of general product L-functions and their applications

  • 期刊论文

  • 发表日期：

    2026年1月22日

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  • Guodong Hua

摘要

Let f be a normalized primitive cusp form of even integral weight for Γ = SL(2, ℤ), and let g be a normalized Hecke–Maass cusp form. In the present paper, for any prescribed integer ℓ ≥ 2, we intend to investigate the average estimates of the Fourier coefficients λf⊗f⊗…⊗ℓf⊗g(n) of the (ℓ+1)-fold product L-functions L(f ⊗ f ⊗…⊗ℓ f ⊗ g, s) involving f and g, where f ⊗ f ⊗…⊗ℓ f ⊗ g is the (ℓ+1)-fold product associated with f and g. As a direct application, we also derive quantitative results for the sign changes of the sequence {λf⊗f⊗…⊗ℓ f⊗g(n)}n≥1 in short intervals, with the indices supported at the positive integers and certain binary quadratic forms.